How do you determine if #f(x) = 2^x + 2^-x# is an even or odd function?

1 Answer
Apr 1, 2016

#f(x)=2^x+2^(-x)# is an even function;
but not an odd function.

Explanation:

even function definition
#f(x)# is an even function if #color(black)(f(x)=f(-x))# for all #x# in the domain of #f#.

#f(x)=2^color(red)(x)+2^color(blue)(-x)#
#f(-x)=2^color(red)(-x)+2^color(blue)(-(-x))=color(blue)(2^(x)+2^color(red)(-x)#
#f(x)=f(-x) rArr f(x)# is even.

odd function definition
#f(x)# is and odd function if #color(black)(f(-x)=-f(x)# for all #x# in the domain of #f#.

if #x=1#
#f(-x=-1)=2^(-1)+2^-(-1)=1/2+2 =2 1/2#
#-f(x=1) = -(2^1+2^-1) = -2 1/2#
#f(-x)!=-f(x)# (when #x=1#) #rArr f(x)# is not an odd function